What is 21.19285285285... as a Fraction?

21.19285 equals 35286116650 as a fraction.

How do you turn 21.19285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.19285 repeating into a fraction, begin writing this simple equation:

n = 21.19285 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (285) and 2 digits in the non-repeating part (19). Multiply both sides by 10⁵ = 100000.

100000 × n = 2119285.285 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

100000 × n = 2119285.285
   100 × n = 21.19285
 99900 × n = 2117166

211716699900 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 6 (the GCF - greatest common factor).

n = 211716699900 = 2117166 ÷ 699900 ÷ 6 = 35286116650. So,

21.19285 = 35286116650 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 35286116650 is also equal to 21321116650 when expressed as a mixed number.

The repeating decimal 21.19285 (vinculum notation) has a repeated block length of 3. It is also represented as 21.19285285285... (ellipsis notation) which equals approximately 21.19285285285285285 (decimal approximation)(*).

The recurring decimal 21.19285 can be written as a ratio of two integers having 352861 as the numerator and 16650 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.